$\begingroup$Oops forgot to mention that, yes generally principal argument is taken as $(-\pi, \pi]$ so you can say for quadrants $1,2$ its measured anticlockwise and for $3,4$ its measured clockwise from $+x$ axis.$\endgroup$
– samjoeJan 6 '18 at 13:21

A good approach to solving this question is to think about the problem geometrically.

The complex numbers $z$ satisfying the condition $|z - 25i| \leqslant 15$ is the region in the Argand plane lying on and inside a circle of radius 15 units centred at $(0,25)$.

The complex number $z$ satisfying the condition $|z - 25i| \leqslant 15$ having the least argument will geometrically be the point on the circle in the first quadrant whose tangent passes through the origin. Let us call this point $z_{\rm min}$ with principal argument $\alpha = \text{Arg} (z_{\rm min})$.

From the geometry of the problem, to find $\alpha$ we have a right-angled triangle with hypotenuse of length 25 units (the distance from the origin to the centre of the circle), side adjacent to the angle $\alpha$ of length 15 units (the radius of the circle), and opposite side of length 20 units as can be found from Pythagoras' theorem.